Identifiability of Kronecker-structured Dictionaries for Tensor Data

TL;DR

This paper establishes conditions under which Kronecker-structured dictionaries for tensor data can be locally recovered from observations, providing bounds on sample complexity and error guarantees.

Contribution

It introduces sufficient conditions for the local recovery of coordinate dictionaries in Kronecker-structured models for tensor data, including sample complexity bounds.

Findings

Sample complexity for dictionary recovery is $ ext{O}(m_k p_k^3 ext{error}^{-2})$.

Recovery guarantees hold with high probability under specified conditions.

Provides theoretical bounds on estimation error for tensor dictionary learning.

Abstract

This paper derives sufficient conditions for local recovery of coordinate dictionaries comprising a Kronecker-structured dictionary that is used for representing $K$th-order tensor data. Tensor observations are assumed to be generated from a Kronecker-structured dictionary multiplied by sparse coefficient tensors that follow the separable sparsity model. This work provides sufficient conditions on the underlying coordinate dictionaries, coefficient and noise distributions, and number of samples that guarantee recovery of the individual coordinate dictionaries up to a specified error, as a local minimum of the objective function, with high probability. In particular, the sample complexity to recover $K$ coordinate dictionaries with dimensions $m_k \times p_k$ up to estimation error $\varepsilon_k$ is shown to be $\max_{k \in [K]}\mathcal{O}(m_kp_k^3\varepsilon_k^{-2})$.